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- W2973593676 abstract "Abstract In this paper, we prove the general solution and generalized Hyers–Ulam stability of n -dimensional functional equations of the form <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mi>i</m:mi> <m:mo>≠</m:mo> <m:mi>j</m:mi> <m:mo>≠</m:mo> <m:mi>k</m:mi> </m:mrow> </m:mtd> </m:mtr> </m:mtable> <m:mi>n</m:mi> </m:munderover> <m:mrow> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo>-</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mi>l</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mi>l</m:mi> <m:mo>≠</m:mo> <m:mi>i</m:mi> <m:mo>≠</m:mo> <m:mi>j</m:mi> <m:mo>≠</m:mo> <m:mi>k</m:mi> </m:mrow> </m:mtd> </m:mtr> </m:mtable> <m:mi>n</m:mi> </m:munderover> <m:msub> <m:mi>x</m:mi> <m:mi>l</m:mi> </m:msub> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>n</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>-</m:mo> <m:mrow> <m:mn>9</m:mn> <m:mo></m:mo> <m:msup> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>20</m:mn> <m:mo></m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mn>12</m:mn> </m:mrow> <m:mn>6</m:mn> </m:mfrac> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:munderover> <m:mrow> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> sum_{begin{subarray}{c}i=1 ineq jneq kend{subarray}}^{n}fbiggl{(}-x_{i}-x_{j}-x_{k}+sum_{% begin{subarray}{c}l=1 lneq ineq jneq kend{subarray}}^{n}x_{l}biggr{)}=biggl{(}frac{n^{3}-9n^{% 2}+20n-12}{6}biggr{)}sum_{i=1}^{n}f(x_{i}), where n is a fixed positive integer with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>ℕ</m:mi> <m:mo>-</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>4</m:mn> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:math> mathbb{N}-{0,1,2,3,4} , in a Banach space via direct and fixed point methods." @default.
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- W2973593676 title "Solution and stability of an n-dimensional functional equation" @default.
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